Multiple time scale derivation of the Fokker-Planck equation for two Brownian spheres suspended in a hard sphere fluid

The Fokker-Planck equation for the distribution function of two massive Brownian spheres, suspended in a fluid of much lighter spheres, is derived from the full hierarchy of exact kinetic equations for the time evolution of the full system consisting of two Brownian and N fluid spheres. The separation of time scales is automatically achieved by a systematic multiple time-scale analysis of the expansion in powers of the square root of the fluid-to-Brownian particle mass ratio. This procedure guarantees uniform convergence of the expansion and requires no extra physical assumptions to justify the separation of time scales. An exact expression is obtained for the mutual friction tensors, which naturally split into a static (Enskog) part and a contribution due to dynamical correlations. The present derivation of the two-particle Fokker-Planck equation also leads to an expression for the fluid-induced, effective depletion force between two Brownian particles.